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anonymous
 5 years ago
solve the first order differential equation: yx(dy/dx)=(dy/dx)y^2e^y
anonymous
 5 years ago
solve the first order differential equation: yx(dy/dx)=(dy/dx)y^2e^y

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Rearrange the equation to put it into a standard form,\[y(x^2+y^2e^y)\frac{dy}{dx}=0\]which is in the form, \[M(x,y)+N(x,y)\frac{dy}{dx}=0\]We check to see if it's exact. We need to ensure,\[\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x} \rightarrow 1=1\]which is true. Now,

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0the potential function,\[\Psi(x,y)\]has that\[\frac{\partial \Psi}{\partial x}=M=y\]and\[\frac{\partial N}{\partial x}=N=x^2y^2e^y\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Integrate the first partial:\[\Psi(x,y)=xy+c(y)\]and then take the partial derivative of this result with respect to y,\[\frac{\partial \Psi}{\partial y}=x+c'(y)\]so that we can identify it with \[\frac{\partial \Psi}{\partial y}=x^2y^2e^y\]

nowhereman
 5 years ago
Best ResponseYou've already chosen the best response.0Above you were saying \[\frac{∂N}{∂x} = 1\] while indeed it is \[2x\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0(note a mistake above dN/dx should be dPsi/dy)

nowhereman
 5 years ago
Best ResponseYou've already chosen the best response.0I mean where you say "We check to see if it's exact."

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Oh  thanks nowhereman  typo with x^2...it should just be x

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0great  now the whole thing's polluted.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[\frac{\partial \Psi}{\partial y}=(x+ye^y)=x+c'(y)\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so that\[c'(y)=2xye^y \rightarrow c(y)=2xye^y(y^22y+2)\]where the last integral is doable using a few integration by parts. Sub. this back in for our solution Psi to give,\[\Psi(x,y)=xy2xye^y(y^22y+2)\]\[=xye^y(y^22y+2)+c=K\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0where c is the constant of integration, and K is a constant due to the fact that this function Psi satisfies\[\frac{d \Psi(x,y)}{dx}=\frac{\partial \Psi}{\partial x}\frac{dx}{dx}+\frac{\partial \Psi}{\partial y}\frac{dy}{dx}=\Psi_{x}+\Psi_y \frac{dy}{dx}=0\]which is the form of the equation we had to solve in the beginning.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Are you seeing any mistakes, nowhereman?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So \[\Psi(x,y)=xye^y(y^22y+2)=c\]for c, some constant.
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